In existing semi-supervised learning frameworks based on VAEs, a prerequisite is learning an informative predictor model based on data with observed labels.
However, in the standard adversarial fair representation learning method, the discriminator is deceived by the feature encoder and it cannot effectively predict the sensitive attributes from feature representations, thereby the vanilla adversarial fair representation learning paradigm is not compatible with semi-supervised VAE.
Motivated by [1]}, we propose to incorporate the decomposed adversarial learning framework into semi-supervised VAE.
As shown in Fig. REF , the decomposed adversarial learning framework has a bias-aware model to learn bias-aware feature representations that mainly capture bias information on sensitive attributes, and a bias-free model to learn bias-free representations that encode bias-independent data information.
We denote the bias-free feature representation as \(\mathbf {r}_f\)  and the bias-aware feature representation as \(\mathbf {r}_b\) .
To encourage the bias-aware model to maximally capture the bias information on sensitive attributes, an attribute predictor is used to infer sensitive attributes labels \(\hat{\mathbf {z}}\)  from the hidden representations learned by the bias-aware model.
We apply an attribute prediction loss \(\mathcal {L}_P\)  to the predictor to optimize its ability in inferring sensitive attributes, which is formulated as:
\(\mathcal {L}_P=-\sum _i \mathbf {z}_i\log (\hat{\mathbf {z}}_i),\) 
